If $I \propto V$, then why is $R = V/I$ and not $I/V$?

Question

I know that the current flowing through a conductor is directly proportional to the potential difference across its ends (by Ohm's Law). Hence,

• I ∝ V
• V ∝ I
• R = V/I, where R is a constant (Resistance)

But why can't it be derived this way?

• I ∝ V
• I = RV
• R = I/V

Won't these two derivations contradict each other?

Thank you

Answer 1

It is true that $V/I$ is a constant for resistors, and also that $I/V$ is a constant. But, of course, they are not the same constant.

$R=V/I$ gives the resistance of a resistor, while $G=I/V$ gives the less commonly-used conductance of a resistor.

Neither of these is a "derivation" of the resistance- $R=V/I$ is a definition of resistance, and you can use Ohm's law to prove that it is constant.

Answer 2

Chris' answer is perfectly right, but I think it's missing a key point: try it out!

Suppose you're the one working with Ohm for his formula (but you have today's equipment for simplicity's sake). You do your reasoning, and you arrive at the two conclusions you pointed out. What to do? You try them out. Build a circuit, turn up the voltage. If you want the circuit to work and not blow up, do you need a more resistant or conducting material? Now do the same with current, and give the proper names to the two quantities you discovered.